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Convergence analysis of a numericalscheme for a tumour growth model
Gopikrishnan C. Remesan
IITB - Monash Research Academy
Monash Workshop on Numerical Differential Equations and Applications2020
Joint work with A/Prof J. Droniou (Monash Uni.) and Prof N. Nataraj (IIT Bombay)
February 11, 20201 / 26 Numerical solutions of free boundary problems
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Contents
1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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Model of tumour–growth
cross-section of tumour spheroid
Assumptions
Cells and fluid exchange matter via the processes, celldivision and cell death.Mass and momentum are conserved internally.No blood vessels. Limiting nutrient - Oxygen, followsdiffusion.
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Model of tumour growth
Domain − 0 < t < T , x ∈ Ω(t) = (0, ℓ(t)).ℓ(t) − tumour length, x = 0 − tumour centre.α − volume fraction of tumour cells, u − cell velocity, c − oxygentension.
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Model of tumour growth
Domain − 0 < t < T , x ∈ Ω(t) = (0, ℓ(t)).ℓ(t) − tumour length, x = 0 − tumour centre.α − volume fraction of tumour cells, u − cell velocity, c − oxygentension.
cell volume fraction (hyperbolic conservation law)
∂α
∂t+ ∂
∂x(αu) =(1+s1)cα(1− α)
1+s1c︸ ︷︷ ︸Birth rate
− s2 +s3c
1+s4cα︸ ︷︷ ︸
Death rate
,
α(0,x) = α0(x).
1+(1/s1), s2 − maximal birth and death rates, s3/s4 − minimaldeath rate.Set f(α, c) = (1+s1)(1−α)c
1+s1c − s2+s3c1+s4c
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Model of tumour growth
Domain − 0 < t < T , x ∈ Ω(t) = (0, ℓ(t)).ℓ(t) − tumour length, x = 0 − tumour centre.α − volume fraction of tumour cells, u − cell velocity, c − oxygentension.
cell velocity (elliptic)
kαu
1− α−µ
∂
∂x
(α
∂u
∂x
)=− ∂
∂x(αH (α)),
u(t,0) = 0, µ∂u
∂x(t, ℓ(t)) = H (α(t, ℓ(t))).
µ − coefficient of viscosity of cell phase. k − interfacial dragcoefficient.Set H (α) = (α −α∗)+/(1− α)2, a+ = max(a,0).
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Model of tumour growth
Domain − 0 < t < T , x ∈ Ω(t) = (0, ℓ(t)).ℓ(t) − tumour length, x = 0 − tumour centre.α − volume fraction of tumour cells, u − cell velocity, c − oxygentension.
Oxygen tension (parabolic)
∂c
∂t− ∂2c
∂x2 = −Qαc,︸ ︷︷ ︸Ox. consumption rate
c(0,x) = c0(x), ∂c
∂x(t,0) = 0, c(t, ℓ(t)) = 1.
Q − Maximum oxygen consumption rate.
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Model of tumour growth
Domain − 0 < t < T , x ∈ Ω(t) = (0, ℓ(t)).ℓ(t) − tumour length, x = 0 − tumour centre.α − volume fraction of tumour cells, u − cell velocity, c − oxygentension.
boundary evolution
ℓ′(t) = u(t, ℓ(t)),
ℓ(0) = 1.
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Idea of extended model
ℓ(0) ℓm0
space (x)
time(t)
T
(α, u, c)
DT
ℓ(t)
α > 0
ℓ(0) ℓm0
space (x)tim
e(t)
T
DT
∂tα+ ∂x(uα) = αf(α, c)
α > 0 α = 0
DT \DT
u|DT \DT= 0
cDT \DT= 1
original model extended model
ℓ as the interface between α > 0 and α = 0.velocity and oxygen tension extended by 0 and 1, respectively.
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Idea of threshold model
ℓ(0) ℓm0
space (x)
time(t)
T
DT
∂tα+ ∂x(uα) = αf(α, c)
α ≤ αthr
DT \DT
u|DT \DT= 0
c|DT \DT= 1
ℓ(0) ℓm0
space (x)
time(t)
T
DT
∂tα+ ∂x(uα) = αf(α, c)
α > 0 α = 0
DT \DT
u|DT \DT= 0
c|DT \DT= 1
extended model threshold model
ℓ as the interface between α > 0 and α <= αthr.velocity and oxygen tension extended by 0 and 1, resp.αthr facilitates estimates on cell velocity and is requirednumerically.
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Threshold solution
A threshold solution (with threshold αthr ∈ (0,1)) and domain DthrT of
the threshold model in DT is a 4-tuple (α,u,c,Ω) such that:0 < m11 ≤ α|Ω(t) ≤ m12 < 1 for all t ∈ [0,T ],m11 ≤ m01, m12 ≥ m02
c ≥ 0,and the following hold:
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Threshold solution
cell volume fraction
The volume fraction α ∈ L∞(DT ) is such that ∀φ ∈ C ∞c ([0,T )×
(0, ℓm)),∫DT
(α,uα) ·∇t,xφdtdx+∫
Ω(0)φ(0,x)α0 dx
+∫
DT
(α −αthr)+ f(α,c)dx = 0.
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Threshold solution
tumour boundary
The set DthrT is of the form
DthrT = ∪0<t<T (t×Ω(t)),
where Ω(t) = (0, ℓ(t)), and we have α ≤ αthr on DT \DthrT .
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Threshold solution
cell velocity
H1,u∂x (Dthr
T ) := v ∈ L2(DthrT ) : ∂xv ∈ L2(Dthr
T )and v(t,0) = 0 ∀ t ∈ (0,T ).
u ∈ H1,u∂x (Dthr
T ) and ∀v ∈ H1,u∂x (Dthr
T ), satisfies∫ T
0at(u(t, ·),v(t, ·))dt =
∫ T
0Lt(v(t, ·))dt, (1)
where at : H1(Ω(t)) × H1(Ω(t)) → R is a bilinear form and Lt :H1(Ω(t)) → R is a linear form as follows:
at(u,v)= k
(α
1−αu,v
)Ω(t)
+µ(α∂xu,∂xv)Ω(t) and (2)
Lt(v)= (H (α),∂xv)Ω(t) . (3)
Extend u to DT by setting u|DT \Dthr
T
:= 0.
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Threshold solution
oxygen tension
H1,c∂x (Dthr
T ) := v ∈ L2(DthrT ) : ∂xv ∈ L2(Dthr
T )and v(t, ℓ(t)) = 0 ∀ t ∈ (0,T ).
c−1 ∈ H1,c∂x (Dthr
T ) satisfies,
−∫
DthrT
c∂tv dxdt+λ
∫Dthr
T
∂xc∂xv dxdt−∫
Ω(0)c0(x)v(0,x)dx
−Q
∫Dthr
T
αcv dxdt = 0, (1)
∀v ∈ H1,c∂x (Dthr
T ) such that ∂tv ∈ L2(DthrT ). Extend c to DT by
setting c|DT \Dthr
T
:= 1.
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Threshold value - comments
To obtain a lower bound strictly greater than zero for α.Facilitates bounded variation estimates on α.To obtain supremum norm bounds on u and ∂xu.
An unavoidable disadvantageResidual volume fraction - creates spurious growth outside thetumour domain.Essential from numerical vantage point.Modified source term (α −αthr)+f(α,c) eliminates spuriousgrowth.As αthr → 0, (α −αthr)+f(α,c) approaches αf(α,c).
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Threshold value - comments
To obtain a lower bound strictly greater than zero for α.Facilitates bounded variation estimates on α.To obtain supremum norm bounds on u and ∂xu.
An unavoidable disadvantageResidual volume fraction - creates spurious growth outside thetumour domain.Essential from numerical vantage point.Modified source term (α −αthr)+f(α,c) eliminates spuriousgrowth.As αthr → 0, (α −αthr)+f(α,c) approaches αf(α,c).
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1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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Discrete scheme
Space: 0 = x0 < · · · < xJ = ℓm, Time: 0 = t0 < · · · tn = T
Uniform discretisation: δ = tn+1 − tn, h = xj+1 −xj .scheme
volume fraction: αnh - upwind finite volume scheme.
Set ℓnh = minxj : αn
j < αthr on (xj , ℓm) andΩn
h := (0, ℓnh).
Conforming Lagrange P1-FEM to obtain unh|Ωn
h, and set
unh = 0 outside Ωn
h.Time-implicit mass lumped P1-FEM to obtain cn
h|Ωnh
, andset cn
h = 1 outside Ωnh.
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Discrete solution
Definition (Time-reconstruct)For a family of functions (fn
h )0≤n<N on a set X, define thetime-reconstruct fh,δ : (0,T )×X → R as fh,δ := fn
h on [tn, tn+1) for0 ≤ n < N .
Definition (Discrete solution)
The 4-tuple (αh,δ,uh,δ, ch,δ, ℓh,δ), where αh,δ, uh,δ, ch,δ, and ℓh,δ arethe respective time-reconstructs corresponding to the families(αn
h)n, (unh)n, (cn
h)n, and (ℓnh)n is called the discrete solution.
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Discrete solution
Definition (Time-reconstruct)For a family of functions (fn
h )0≤n<N on a set X, define thetime-reconstruct fh,δ : (0,T )×X → R as fh,δ := fn
h on [tn, tn+1) for0 ≤ n < N .
Definition (Discrete solution)
The 4-tuple (αh,δ,uh,δ, ch,δ, ℓh,δ), where αh,δ, uh,δ, ch,δ, and ℓh,δ arethe respective time-reconstructs corresponding to the families(αn
h)n, (unh)n, (cn
h)n, and (ℓnh)n is called the discrete solution.
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Why a mixed numerical scheme?Finite volume method
Respects mass conservation property at the discrete level.Upwind flux (+ CFL) yields a stable scheme.FVM - significant numerical diffusion.Large error in locating ℓn
h as the boundary where αnh becomes 0.
Solution: Locate ℓnh as minxj : αn
h < αthr on (xj , ℓm].
ℓm x|
αth
r
α(t, ·)
αnh
ℓnh
αnh < αthrΩn
h
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Why a mixed numerical scheme?
Finite element methodsVelocity equation - elliptic, oxygen tension equation - parabolic.Unknown Lagrange P1-FEM - boundary nodes of (xj ,xj+1), andeasy to compute the upwind flux.Mass lumping in oxygen tension equation is crucial to obtain L∞
bounds.Time-implicitness yields stability in L2(0,T ;H1(0, ℓm)).
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1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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uh,δ: continuous modification of uh,δ
||ℓm
uh,δ(t,·)
ℓh,δ(t, ·)
(a) uh,δ(t, ·).
ℓmℓh,δ(t, ·)
uh,δ(t,·)
||
uh,δ(t, ℓh,δ(t))
(b) uh,δ(t, ·).
Figure: The left-hand side plot illustrates the discontinuous function uh,δ
and the right-hand side plot illustrates the continuous modification uh,δ.
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Few notationsMass lumping operator
Set χj = (xj −h/2,xj +h/2), Sh,ML − piecewise constantfunctions on χj .Mass lumping operator: Πh : C 0([0,L]) → Sh,ML suchthat Πhw =
∑Jj=0 w(xj)1Xj
.
Set Πh,δch,δ by Πh,δch,δ(t, ·) := Πh(ch,δ(t, ·)).
b b| | |xj xj+1xj−1/2 xj+1/2 xj+3/2
w(x)
x
w(xj)w(xj+1)
w(x0)
Πhww
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Few notations
Time-dependent spaces
L2c(0,T ;H1(0, ℓm)) := f ∈ L2(0,T ;H1(0, ℓm)) : f(t, ℓ(t)) = 0
for a.e. t ∈ [0,T ],
L2u(0,T ;H1(0, ℓm)) := f ∈ L2(0,T ;H1(0, ℓm)) : f(t,0) = 0
for a.e. t ∈ [0,T ].
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Main Theorem I - Hypothesis
Theorem (compactness)
Let the properties stated below be true.The initial volume fraction α0 belongs to BV (0, ℓm) and0 < m01 ≤ α0 ≤ m02 < 1, where m01 and m02 are constants.The discretisation parameters h and δ satisfy the followingconditions:
ρCCF L ≤ δ
h≤ CCF L :=
√a∗µ
2ℓm
|1−a∗|2
|a∗ −α∗|and
δ < min(
1−ρ
s2,
2(1−ρ)1+s2
),
where ρ, a∗ and a∗ are constants chosen such that ρ < 1,0 < a∗ < m01, and 0 < m02 < a∗.
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Main Theorem I - Conclusions
Theorem (compactness)Then, there exists a finite time T∗ = T∗(ρ, a∗,a∗), a subsequence of thefamily of functions (αh,δ, uh,δ, ch,δ, ℓh,δ)h,δ and a 4-tuple offunctions (α, u,c,ℓ) such that
α ∈ BV (DT∗)c ∈ L2
c(0,T∗;H1(0, ℓm))u ∈ L2
u(0,T∗;H1(0, ℓm))ℓ ∈ BV (0,T∗)
with DT∗ = (0,T∗)× (0, ℓm) and as h, δ → 0,αh,δ → α almost everywhere and in L∞ weak⋆ on DT∗ ,Πh,δch,δ → c strongly in L2(DT∗) and ∂xch,δ ∂xc weakly inL2(DT∗),uh,δ u and ∂xuh,δ ∂xu weakly in L2(DT∗), andℓh,δ → ℓ almost everywhere in (0,T∗).
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Main Theorem II
Theorem (convergence)Let (α, u,c,ℓ) be the limit provided by the compactness theorem.Define Ω(t) := (0, ℓ(t)) and the threshold domain
DthrT∗ := (t,x) : x < ℓ(t), t ∈ (0,T∗)
and let u := u on DthrT∗ and u := 0 on DT∗\Dthr
T∗ . Then, (α,u,c,Ω) is athreshold solution with T = T∗.
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Idea for proof of Main Theorem I
Proof is through inductive arguments on time-step, n.Fix two constants a∗ ∈ (max(α∗,m02),1) anda∗ ∈ (0,min(αthr,m01)).The time of existence T∗ on a∗ and a∗, and is explicitly providedby Theorem (well-posedness).
Theorem (well-posedness)For all n ∈ N such that tn ≤ T∗, αh,δ(tn, ·), uh,δ(tn, ·), and ch,δ(tn, ·)are well defined, and it holds:
a∗ < αh,δ(tn, ·)|Ωnh
< a∗,0 ≤ ch,δ(tn, ·)|(0,ℓm) ≤ 1.
Necessary compactness results proved using supremum normbounds from the Theorem (well-posedness).
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Proof of well-posedness TheoremStep 1: Energy estimate of un
h
There exists a unique solution unh to discrete weak form of ve-
locity equation in Ωnh and it satisfies the following estimates:∣∣∣∣∣∣∣∣√αh,δ(tn, ·)∂xun
h
∣∣∣∣∣∣∣∣0,Ωn
h
≤√
ℓm|a∗ −α∗|µ|1−a∗|2
and∣∣∣∣∣∣∣∣∣∣
√αh,δ(tn, ·)un
h√1−αh,δ(tn, ·)
∣∣∣∣∣∣∣∣∣∣0,Ωn
h
≤
√ℓm
kµ
|a∗ −α∗||1−a∗|2
.
Keeping the coefficients yields optimal estimates, which improvesthe existence time T∗.Estimate on ∂xun
h yields
||uh,δ(tn, ·)||L∞(0,ℓm) ≤ ℓm√a∗µ
|a∗ −α∗||1−a∗|2
. (3)
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Proof of well-posedness Theorem
Step 3: BV and L∞ bound on ∂xuh,δ(tn, ·)
It holds:
||µαh,δ(tn, ·)∂xuh,δ(tn, ·)−H (αh,δ(tn, ·))||BV (0,ℓm)
≤ ℓm
√k
µ
|a∗ −α∗||1−a∗|5/2 ,
||(µαh,δ(tn, ·)∂xuh,δ(tn, ·))−||L∞(0,ℓm) ≤ ℓm
√k
µ
|a∗ −α∗||1−a∗|5/2 , and
||µαh,δ(tn, ·)∂xuh,δ(tn, ·)||L∞(0,ℓm) ≤ ℓm
√k
µ
|a∗ −α∗||1−a∗|5/2
+a∗(a∗ −α∗)(1−a∗)2 .
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Proof of well-posedness Theorem
Step 3: L∞ bound on αh,δ(tn, ·)
There exists T∗ > 0 such that if n+1 ≤ N∗ := T∗/δ, then
a∗ ≤ minj :xj∈Ωn+1
h
αn+1j ≤ max
0≤j≤J−1αn+1
j ≤ a∗.
Step 4: L∞ bound on ch,δ(tn, ·)
The discrete weak form corresponding to the oxygen tensionequation has a unique solution cn+1
h in Ωnh, and it holds 0 ≤
cn+1h ≤ 1.
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1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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Check-list of compactness estimates
1 a uniform L2(0,T∗;H1(0, ℓm)) estimate for the family ch,δh,δ −weak L2(0,T∗;H1(0, ℓm)) convergence.
2 a uniform spatial and temporal BV estimate for the familyαh,δh,δ − strong Lp(DT∗) convergence.
3 a uniform BV estimate for the family ℓh,δh,δ − strongLp(0,T∗) convergence.
4 the family Πh,δch,δh,δ is relatively compact in L2(DT∗) −strong L2(DT∗) convergence.
5 a uniform L2(0,T∗;H1(0, ℓm)) estimate for the familyuh,δh,δ − weak L2(0,T∗;H1(0, ℓm)) convergence .
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Relative compactness of Πhch,δh,δ
DefinitionAuxiliary function Define ch,δ := ch,δ −1. For a fixed ϵ > 0, define theauxiliary function φn
h,ϵ : [0, ℓm] → [0,1] by
φnh,ϵ(x) =
1 0 ≤ x ≤ ℓnh − ϵ,
(ℓnh −x)/ϵ ℓn
h − ϵ < x ≤ ℓnh,
0 ℓnh < x ≤ ℓm.
ℓmℓnhℓnh − ϵ
φnh,ϵ
1
0 b
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The mass lumped function can be split into
Πh,δ ch,δ = Πh,δ(ch,δφh,ϵ)+Πh,δ(ch,δ(1−φh,ϵ)),
where φh,ϵ = φnh,ϵ on [tn, tn+1) for 0 ≤ n ≤ N∗ −1.
The second term can be bounded by:
||Πh,δ(ch,δ(1−φh,ϵ))||L2(DT∗ ) ≤√
T∗ϵ.
TheoremThe family of functions Πh,δ(φh,ϵch,δ)h,δ is relatively compact inL2(DT∗).
Proof follows from the Discrete Aubin - Simon Theorem.
TheoremThe family of functions Πh,δch,δh,δ is relatively compact inL2(DT∗).
Proof follows from the fact ϵ > 0,
Πh,δ ch,δh,δ ⊂ Πh,δ(φh,ϵch,δ)h,δ +BL2(DT∗ )
(0;
√T∗ϵ
).
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1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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Idea of proof of convergence theorem
1 The domains
Ah,δ := (t,x) : x < ℓh,δ(t), t ∈ (0,T∗)
converge to
DthrT∗ := (t,x) : x < ℓ(t), t ∈ (0,T∗).
2 The limit function α satisfies the weak form (volume fraction)with T = T∗.
3 The restricted limit function u|DthrT∗
satisfies weak form (cellvelocity) with T = T∗.
4 The limit function c|DthrT∗
satisfies weak form (oxygen tension)with T = T∗.
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1 Model
2 Discretisation
3 Main Theorem
4 Compactness results
5 Convergence results
6 Numerical results
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Parameters
k = 1, µ = 1, Q = 0.5, s1 = 10 = s4, s2 = 0.5 = s3, α∗ = 0.81.The bounds of the cell volume fraction are set to be a∗ = 0.4 anda∗ = 0.82.The extended domain length ℓm is set as 10.The threshold value is taken as αthr = 0.1.ρ = 0.1, δ = 1E−3 and h = 5E−2.Set T∗ = 50.Predicted time by compactness theorem: 1E−7 to 1E−1.
1Breward, C.J.W., Byrne, H.M. and Lewis, C.E., 2002. The role of cell-cellinteractions in a two-phase model for avascular tumour growth. J. of Math. Bio.,45(2), pp. 125-152.
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0 2 4 60
0.2
0.4
0.6
0.8
x
αh,δ(t,x)
t = 5t = 10t = 15t = 20t = 25t = 30t = 35t = 40t = 45t = 50
(a) cell volume fraction
0 2 4 6
0
5
·10−2
x
uh,δ(t,x)
t = 5t = 10t = 15t = 20t = 25t = 30t = 35t = 40t = 45t = 50
(b) cell velocity
0 2 4 60
0.2
0.4
0.6
0.8
1
x
c h,δ(t,x)
t = 5t = 10t = 15t = 20t = 25t = 30t = 35t = 40t = 45t = 50
(c) oxygen tension
0 10 20 30 40 500
2
4
6
t
` h,δ(t)
(d) tumour radius
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Conclusive remarks
Sufficiency of compactness and convergence theorems - Existenceof solutions beyond T∗ is possible.Convergence theorem guarantees existences of a domain D
αthrT .
However, it is not known whether it is unique.Framework can be extended to similar problems and models.Higher dimensional study (on going work).
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Conclusive remarks
Sufficiency of compactness and convergence theorems - Existenceof solutions beyond T∗ is possible.Convergence theorem guarantees existences of a domain D
αthrT .
However, it is not known whether it is unique.Framework can be extended to similar problems and models.Higher dimensional study (on going work).
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Conclusive remarks
Sufficiency of compactness and convergence theorems - Existenceof solutions beyond T∗ is possible.Convergence theorem guarantees existences of a domain D
αthrT .
However, it is not known whether it is unique.Framework can be extended to similar problems and models.Higher dimensional study (on going work).
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Conclusive remarks
Sufficiency of compactness and convergence theorems - Existenceof solutions beyond T∗ is possible.Convergence theorem guarantees existences of a domain D
αthrT .
However, it is not known whether it is unique.Framework can be extended to similar problems and models.Higher dimensional study (on going work).
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Thanks
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